rollo sara k

8/9/2019 Rollo Sara K

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c2009

SARA KATHERINE ROLLO

ALL RIGHTS RESERVED


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INTEGRATING HEALTH EDUCATION AND LEISURE TIME INTO

ECONOMIC GROWTH MODELING

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Sara Katherine Rollo

December, 2009


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INTEGRATING HEALTH EDUCATION AND LEISURE TIME INTO

ECONOMIC GROWTH MODELING

Sara Katherine Rollo

Thesis

Approved:

AdvisorDr. Gerald W. Young

Co-AdvisorDr. Curtis Clemons

Faculty Reader

Dr. Timothy S. Norfolk

Department ChairDr. Joseph Wilder

Accepted:

Dean of the CollegeDr. Chand Midha

Dean of the Graduate SchoolDr. George R. Newkome

Date

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ABSTRACT

This paper models the economic growth of a developing country. An integrated

approach incorporating, capital, technology, and human capital is used. The model is

a Hamiltonian problem that maximizes utility by determining the amount of time that

should be spent each day in production, investment in human capital, investment in

health education, and leisure. Adding leisure time into the model is used to determine

if this will allow economic growth of the country. Numerical solutions of the model

are found and show that adding leisure increases productivity and technology, but

results in a decrease to life expectancy. Stability analysis of the model shows that

leisure provides multiple unstable paths for the country, although a trap stable fixed

point can still occur.

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ACKNOWLEDGEMENTS

I would like to thank Dr. Young and Dr. Celmons for all of their help on this project.

I would also like to thank Dr. Sheppard for her help and insight into the economic

background for this project. Thank you to The University of Akron for the use of

its facilities. Thank you to Joe Tucker, also, who started this research before me

and without his idea, my research would not have been possible. Thank you to Dr.

Norfolk also for reading my thesis.

I want to say a special thanks to my family and friends who have prayed for

me throughout working on my thesis, without which I wouldnt have finished.

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TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

CHAPTER

I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Recovering Tucker . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Parameter List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

II. CONSTRUCTING THE ORIGINAL MODEL . . . . . . . . . . . . . 15

2.1 The Original Model . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Interpretation of Technology and Life Expectancy Equations . . . 22

2.3 Solving the Original Model . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Interpretations of the Original Model . . . . . . . . . . . . . . . . 28

III. STEADY STATE OF THE ORIGINAL MODEL . . . . . . . . . . . 31

3.1 Steady State Solutions . . . . . . . . . . . . . . . . . . . . . . . . 31

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3.2 Results of the Original Model . . . . . . . . . . . . . . . . . . . . . 34

IV. CONSTRUCTING THE TRANSFORMED MODEL . . . . . . . . . 37

4.1 The Transformed Model . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Finding the Fixed Points . . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Results of the Transformed Model . . . . . . . . . . . . . . . . . . 44

V. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

APPENDIX A. RECOVERING TUCKER . . . . . . . . . . . . . . . . . 60

APPENDIX B. HAMILTONIAN DERIVATIONS . . . . . . . . . . . . . 62

APPENDIX C. STEADY STATE DERIVATIONS . . . . . . . . . . . . . 72

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LIST OF TABLES

Table Page

1.1 Definition of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Definition of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.1 Equilibrium Values for the Original Model . . . . . . . . . . . . . . . . 36

3.2 Eigenvalues and Eigenvectors for the Original Model . . . . . . . . . . . 36

4.1 Eigenvalues and Eigenvectors for the Transformed Model . . . . . . . . 52

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LIST OF FIGURES

Figure Page

2.1 Technology Equation (2.2): Varying , the rate of diffusion of tech-nology. Here a = 0.075 and A0 = 0.015. . . . . . . . . . . . . . . . . . . 22

2.2 Life Expectancy Equation (2.4): Varying , the measure of influence

of capital. Here Q = 85 and Q0 = 35. . . . . . . . . . . . . . . . . . . . 23

2.3 Life Expectancy Equation (2.4): Varying , the measure of influenceof time spent in health education. Here Q = 85 and Q0 = 35. . . . . . 24

2.4 Life Expectancy Equation (2.4): Comparing developed versus devel-oping countries with varied , and K. Here Q = 85 and Q0 = 35. . . 25

4.1 a = 0.075 and A0 = 0.1404x103. . . . . . . . . . . . . . . . . . . . . . 44

4.2 Q = 85 and Q0 = 35. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 The fixed points ep, eq and l and the sum of the three versus withtwo values of where Q0 = 35, = 0.00932 and = 0.02. . . . . . . . . 46

4.4 Time spent in production, u, versus with two values of whereQ0 = 35, = 0.00932 and = 0.02. . . . . . . . . . . . . . . . . . . . . 47

4.5 Life expectancy, q, versus with two values of where Q0 = 35, =0.00932 and = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.6 Technology, A, versus with two values of where Q0 = 35, =0.00932 and = 0.02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.7 Ratio of consumption per capital, v, versus with two values of where Q0 = 35, = 0.00932 and = 0.02. . . . . . . . . . . . . . . . . 50

4.8 Ratio of capital per human skill level, x, versus with two valuesof where Q0 = 35, = 0.00932 and = 0.02. . . . . . . . . . . . . . . 51

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4.9 Time spent in health education, eq, versus time, in years to see ifthe stable path can move away from the fixed point. . . . . . . . . . . . 53

4.10 Time spent in leisure, l, versus time, in years to see if the stablepath can move away from the fixed point. . . . . . . . . . . . . . . . . 54

4.11 Ratio of capital per human skill level, x, versus time, in years to seeif the stable path can move away from the fixed point. . . . . . . . . . 54

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CHAPTER I

INTRODUCTION

1.1 Overview

We often hear that The rich get richer and the poor get poorer, but is it really true?

In an article in the New York Times, Conley suggests that this may be true because

the more money you make and the richer you are, the more you want to work to keep

the money you have [1]. In summary of the above-mentioned article, the rich like

to compete against each other, and if one sees that another has money, that drives

him/her to work more and make more money. On the other end of the spectrum,

the poor stay in a stable state because they have no one with whom to compete.

Therefore, the poor just stay poor, while the rich really do get richer [1]. Moving to a

larger scale, the same distinction holds true for developed versus developing countries.

Can this be helped? What can be done for these developing countries to get them

out of their economic state, which is causing them to stay poor?

To try and answer these questions, economic models are examined and mod-

ified to fit the economy of a developing country. Different terms are integrated into

an existing model [2] to form a new model.

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After adding such terms, the new model is solved and the governing equations are

interpreted, according to the economy of a developing country.

This Masters Thesis is an extension of Tucker (2008) [2]. Tucker incorporates

life expectancy into a model that integrates technology and human capital. Tuckers

approach is an attempt to validate whether international health aid and education

is an economically viable and profitable endeavor [2]. Tucker discovered that it is

worthwile to include health education into a struggling economy because it does

increase life expectancy of that country. In addition to incorporating life expectancy,

this proposed Masters Thesis model will extend Tuckers approach by adding leisure

into the model. This is included to determine whether having a break in the work

day would make the workers more productive in the long run. By incorporating

leisure, looking at fixed points and varying the parameter that measures leisure, we

find that a country produces more but has a shorter life expectancy as leisure time

increases. After finding the fixed points and doing a linear stability analysis on the

model proposed in this thesis, we see that it is necessary to perform a change of

variables. We also find the fixed points of the newly transformed model and a linear

stability analysis is performed using Maple. The stability analysis shows that there

are multiple unstable paths in which an economy is improving in some way, whether

it be through capital increasing or technology improving. Despite the opportunity for

continued improvement, there is still opportunity for the economy to be trapped on

a stagnant path, although it may be at higher state of living. Future research for this

project would be to break up a persons life into different stages and see when he/she

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is more productive and when he/she starts becoming less productive or completely

unproductive for the economy.

The remainder of this chapter presents a literature survey on existing models

and presents the assumptions. These models are evaluated and explained in later

sections and are used as stepping stones for this proposed model. The model proposed

in this thesis will be presented in Chapter 2 and so will the governing equations and

interpretations. Chapter 3 presents the steady state solutions for this model and

Chapter 4 introduces a change of variables and a transformed model is presented

with its steady state solutions. The results and conclusions are provided in Chapter

5. In the Appendix, the derivations of the model are explained.

1.2 Literature Survey

In [3], Charles Jones reviews the Solow Model, which uses a Cobb-Douglas form

production function. This model treats technology as an exogenous force, which is

growth from an outside source, and treats socio-political stability as a coefficient that

can either help or hinder economic growth for an economy [3].

Jones focuses on how to invest excess money: should it be invested in human

or physical capital? This question leads to the Romer model [4]. Romer focuses

on distributing investments across different areas, such as human capital, research

and development, or physical capital, with a goal of maximizing growth and utility

[4]. These models are good for developed countries, but the assumptions for devel-

oped countries are different from those of developing countries. For instance, life

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expectancies are different for developed and developing countries, which would cause

the economy to invest differently. Therefore, we seek to find a model that would be

a better fit for a developing country, rather than a developed country.

Two models that more accurately relate to a developing country are those by

Fabrizio Zilibotti [5] and Michal Kejak [6]. Both models are derived from Robert

Lucas [7]. Zilibotti uses the Romer and Solow models in his paper. Zilibottis

model separates a region where growth is Solow-type, with convergence to a sta-

tionary steady-state, from a region where growth is Romer-type, with endogenous

self-sustained growth [5]. The logistic equation and its solution are used in Zilibottis

model to relate technology to capital accumulation [5]. The logistic equation and its

solution are:

A

A=

a A

aK, (1.1)

A =a

1 + ( a

A0 1)eK

, (1.2)

where = ddt

. In these equations A is technology, K is aggregate capital, which is

the total capital available throughout the economy. The variable t is time and

is a diffusion parameter that is positive. Many variables and parameters are used

throughout the model. For a concise definition of each variable and parameter used,

refer to Tables 1.1 and 1.2. Notice that a change is made in Zilibottis notation

in order to remain consistent with the notation of the proposed model. What is

important to note here is that as K approaches 0, A approaches A0, which means that

technology is approaching its lowest level. On the other hand, when K approaches

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, A approaches a, which is the highest feasible level of technology in the economy.

We have that K = 0 is the initial condition and, as mentioned, we get A0, the lowest

level of technology. Due to the endogenous growth, as capital increases, technology

grows also. This proposed model uses (1.1) and (1.2).

We use Zilibottis non-linear production function, y,

y = DA(K)k + Z[A(K)k]. (1.3)

Here D and Z are both parameters, k is capital and is a parameter for the non-

linear term associated with production growth. Aggregate capital, K, is independent

of the economys capital k. Therefore, when taking derivatives with respect to k, any

term with K is not affected.

Zilibotti and Daron Acemoglu [8] consider productivity differences among

countries with the same level of technology. It is argued that even if a developed

and a less developed country have the same level of technology, the productivity of

a less developed country will be lacking because they have unskilled workers using

technology designed for skilled workers [8]. Learning how to use the new technology

is essential for having high productivity levels.

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Michal Kejak [6] extends Lucas model [7] by incorporating human capital and

productivity endogenously. The following equations from Kejak are also important

to this papers model:

B

B=

BH B

BHH, (1.4)

B(H) =BH

1 +

BHB0 1

eH

, (1.5)

h = B(H)(1 u)h. (1.6)

In these equations, B0 is the lowest level of productivity, BH is the highest level of

productivity and B is productivity at time t. According to Kejak, is a measure of

the speed of diffusion and H is the average level of human capital in the economy

[6]. As H increases, productivity approaches the highest level of production and as

H decreases, productivity decreases to the lowest level of productivity. Equations

(1.5) and (1.6) will be used in this model. The argument used for capital to delineate

k and K holds true for human skill level h and H. It should also be noted that at

equilibrium, k = K and h = H.

In equation (1.6), (1 u) is the amount of non-leisure time spent investing

in human capital. Kejaks model maximizes economic growth if (1u) is optimized.

This proposed model allots time for production, u, investment in productivity, ep, for

health education, eq and for leisure time in the work day, l. Therefore, we have

u = 1 ep eq l. (1.7)

To incorporate the leisure term, l, we follow [9]. These authors also take

their model from Lucas, but incorporate leisure into the utility function. The utility

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function for this papers model is taken from [9], but is modified with a time dis-

counting factor. Incorporating leisure into the model introduces a potential source

of non-convexities in the optimization problem and leads to the possible existence of

multiple growth paths [9]. The utility function is defined as

U[c(t), l(t)] =[c(t)l(t)1]1

1 . (1.8)

In this utility function, l is leisure, t is time, c is consumption at time t, is a

parameter that measures the value of leisure with 0 < < 1, and is a parameter

which measures risk aversion, 0 < < 1. Ladron, Ortigueira and Santos mention

that there is no previous study on the role of leisure in the process of growth and

they incorporate leisure to determine how time can be allocated between production

of goods, education and leisure throughout the work week [9]. It is found that in those

balanced growth paths, having higher physical capital, individuals will devote more

time to leisure and less time to education. The proposed model will help determine

if adding leisure to the work day will aid with improving the economic state of a

country. We maximize the above utility function.

In [10], Paul A. de Hek also incorporates leisure and consumption into his

utility function, where he looks at the relationship between consumption and leisure.

As found in [11], as the price of one product increases/decreases the demand of the

substitute product will increase/decrease. If Coca-cola and Pepsi are substitutes, the

increase in the price of Pepsi will result in an increased demand for Coca-cola. On the

otherhand, as the price increases/decreases, the demand for a complement product

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decreases/increases. Considering hot dogs and hot dog buns, which are complements,

if the price of hot dogs increase, the demand for hot dog buns will decrease. Looking

at consumption and leisure as goods, Hek proposes that if the two are substitutes, the

model can generate multiple steady states and if they are complements, the optimal

path may turn out to be cyclical [10]. If they are substitutes and an individual has

low-income in one period, he/she is likely to still have low-income in the next period

and same for those with high-income. If consumption and leisure are complements

then an individual who has low-income in one period may work harder in the next

period to have a higher income and an individual with high-income in one period

may have low-income in the next, which results in a cyclical behavior [10]. In [12],

Salvador Ortigueira defines leisure as the amount of time devoted to activities which

are factored by the level of human capital. Individuals derive utility from consumption

and leisure and those having a greater amount of human capital can obtain a higher

level of utility [12].

Using foreign investment as an aid to promote growth is an issue of dispute

and in [13] it is expressed that having capital mobility helps a stagnant country change

course and gives support for foreign investment and growth. Tucker introduces life

expectancy into the model to determine if having a longer life will help improve the

state of the economy, such as improving upon and increasing capital. This issue is

also studied in [14]. A model is built where increasing life expectancy is endogenously

determined by public investment in health and it is determined that a longer life ex-

pectancy results in an increase in savings and a larger workforce [14]. The conclusion

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drawn is the more effective the health expenditure, the bigger the increase in growth

because it reduces mortality, which is a key element in growth [14]. This conclusion

is also supported in [15], where it is shown that health is a complement to growth.

These articles support this model also, which shows that effective health education

does increase life expectancy, which helps economic growth. In [16], the engine of

growth is given by the change in human capital. Human capital depends on the per-

sons decisions on schooling and age of retirement, which affect productivity. A model

is built in which individuals choose education time and retirement age and pensions

depend on the contributions made by workers during their active lives [16], which is a

good start for looking into further research to determine the most productive part in

an individuals life. This model found that individuals retire sooner when they expect

to receive social security and that social security also produces a negative effect on

education [16].

Tuckers model [2] adjusts previous models by incorporating health education

and life expectancy to show the positive economic returns of a healthy society. In

addition to Tucker, we are adding leisure time to determine if a healthy society, that

is also well-rested throughout the work day, will further increase both the economic

growth and the economic returns of a developing country. In other articles regarding

the health of developing countries, it was discovered that AIDS is among the top

causes of economic stagnation [17]. The AIDS epidemic is so real and severe that it

prohibits workers from being as productive as they can be. With health education, it

is hoped that individuals in these economies will not contract AIDS as easily and will

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grow economically. Tucker discovered that teaching about AIDS and other diseases

through health education increases the life expectancy of a developing country.

1.3 Recovering Tucker

This proposed model is an extension of [2]. In order to get the model from [2], certain

limits are taken. If leisure approaches zero and = 1 we get Tuckers model. From

his model, additional limits are taken to retrieve Zilibotti and Kejak. Tuckers model

reduces to Zilibotti, by setting ep, eq and h to zero and q is set as a constant. To

recover Kejaks model from Tuckers model set eq = 0, and = 1. Also one must set

q and A as fixed constants.

1.4 Assumptions

This paper seeks to maximize the utility, in the model proposed in thesis but not the

transformed model, with respect to consumption, leisure, human skill level, capital,

health education and production education in developing countries, so that their

economy is able to leave the stagnant state it is in, and prosper. Production, health

education and leisure make up the three-sector market and we want to determine how

much time should be spent within each sector. The idea is to see if adding leisure

time in the production sector will create more productive workers, thus increasing

overall funding for the economy. A second sector is health education, which creates

more time learning about health and will increase life expectancy. The final sector is

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time spent in human capital, which will yield higher productivity in the production

sector.

This model focuses on the effects of health education and leisure time in the

economy. The model is assumed to be for a developing country, but can also be

used for developed countries so long as the following assumptions are met. There is

a highest life expectancy Q, that is attainable by an individual in a country. The

life expectancy, q is assumed to be initially low for a developing country, because the

life expectancies are not high in such a country. It is also assumed that a person is

productive throughout his/her entire life. Another option for future work would be

to look at the different stages in ones life to see when he/she is more productive and

will be more beneficial to economic growth. This model neglects the factor of stress

throughout a persons life, and adding such a factor to the life expectancy equation

would also be helpful for future work.

1.5 Parameter List

The definitions and explanations of each parameter and variable are given when they

are introduced in an equation; however, for conciseness, a complete list of parameters

and variables are provided in the following tables

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Table 1.1: Definition of Variables

Variable Definition Unit

t Time Years

A Technology as a function of time Unitless

K Aggregate capital Output

k Capital Output

q Life expectancy Years

eq Time spent in health education Unitless

B Productivity level 1/Years

H Aggregate skill level Unitless

h Skill level Unitless

ep Time spent in production education Unitless

l Time spent in leisure Unitless

u Time spent in production Unitless

c Consumption Output/Years

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Table 1.2: Definition of Parameters (cont.)

Parameter Definition Unit

Productivity diffusion Unitless

Capital diffusion 1/Output

Depreciation of capital 1/Years

Measure of risk aversion Unitless

a Highest level of technology Unitless

A0 Lowest level of technology Unitless

Technology diffusion 1/(Years*Output)

Time discount factor 1/Years

r Interest rate 1/Years

Z Non-linear output coefficient Output1/Years

D Linear output coefficient 1/Years

Non-linear output coefficient Unitless

Parameter on leisure Unitless

G Cobb-Douglas multi-factor productivity constant Output/Years

Capital share of production Unitless

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CHAPTER II

CONSTRUCTING THE ORIGINAL MODEL

2.1 The Original Model

This model is similar to Tuckers model, with most of the equations being identical.

Two of the equations in this model are modified versions of Tuckers equations to

accommodate the allotment of leisure time throughout the work day. The equation for

technological growth and its differential equation, the equation for life expectancy and

its differential equation are taken directly from Tuckers model and will be presented

with their respective explanations first. The technological growth equation is

dA

dt=

a A

a Ad

dt[qK]. (2.1)

In this equation, A is the level of technology at time t, and a is the highest level

of technology the world can reach. Also, in this equation, K represents aggregate

capital, which is the total capital available throughout the economy, and q stands for

life expectancy measured in years. Finally, > 0 is the diffusion of change in the

technological level into the economic system. The solution to equation (2.1) is

A(t) =a

1 +

aA0 1

e[q(t)K(t)]

. (2.2)

Here we have A0, which is the lowest level of technology. According to these equations,

if a person lives a longer life, represented by q, he/she has more time to create and

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use technology that is beneficial to the economy [2]. Using the same logic, as capital

increases, technology approaches the highest point of production. Note that as qK

tends to 0, then A tends to A0, the lowest level of technology. The rate at which

an economy or country reaches the highest level of technology, a, depends on the

value of , the rate of diffusion of technology. The parameter also shows how

effective individuals of a country are at using their resources, or capital, and using

their wisdom to increase life expectancy, which, in turn, can improve and increase

technology. With a small q and K, there is not a long enough time to gain enough

experience, nor are there enough resources to increase technology. This is illustrated

in Figure 2.1. As a point of comparison, Kejak does not integrate technology into his

model; therefore, = 0 and A is constant.

It is also important to delve into the life expectancy function, as the changes

in q affect A. The differential equation for life expectancy is

dqdt

= Q qQ

q ddt

[eq + K] , (2.3)

where Q is the highest possible life expectancy. In this equation, eq represents non-

leisure, non-production time spent in health education. The diffusion affect of the

time spent in health education is represented by > 0. The parameter is also

positive and measures the influence of capital on life expectancy. Neither Zilibotti

nor Kejak integrate life expectancy into their models, and hence q would be constant

in each of their models. When this equation is solved, we find that

q(t) =Q

1 +

QQ0 1

e[(eq)+(K(t))]

. (2.4)

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Following the previous logistic functions, Q0 is the lowest feasible life expectancy in

the country. As K and eq increase, q approaches Q, which is the highest possible life

expectancy. As eq tends to 0 and as K tends to 0, q tends to Q0. A small eq implies

there is not much time spent in health education, whereas a large eq implies there is

much time. A small K implies the country is poor and a large K represents a rich

country. The constraints and determine how quickly the economy or country is

able to reach the highest possible life expectancy. The size of , the diffusion effect

of health education, can be thought of as the effectiveness of the teaching. A small

implies non-effective teaching, whereas a large implies very effective teaching.

The same logic applies for , the diffusion effect of capital. A small implies that

a country is non-effective in using resources to improve health. In contrast, a large

implies a country is very effective in using resources to improve health. This is

illustrated through Figures 2.2, 2.3 and 2.4.

Using Equation (2.5) from Tuckers model, the following logistic equation is

found

B(H) =BF

1 +

BFB0 1

eH

. (2.5)

This equation is based on assumptions from Kejak and Lucas regarding human cap-

ital and productivity. The idea is that spending time to improve human capital pro-

motes technical progress [2]. In this equation, BF is the highest level of productivity

throughout the world market. Also, > 0, the speed of diffusion for productivity

determines the ease with which a country can reach its highest feasible level of pro-

ductivity. The lowest level of productivity of a country, which is attained only at the

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lowest level of human capital, is denoted by B0. It should be pointed out that for all

time t, BF B B0. As human capital increases, B approaches BF and as human

capital decreases toward zero, B approaches B0.

The next equation follows the form of Kejak and assumes that the level of

productivity in the education sector depends on the developmental level of a society,

which is expressed by the average level of human capital [6],

dh

dt= B(H)eph. (2.6)

In this equation, which is the constraint on human skill level, B(H) is the level of

productivity in the education sector and h is the level of skill held by the popula-

tion. Here, ep is the proportion of non-leisure time spent in production education, as

opposed to health education. Zilibotti does not consider skill level, hence h = 0 for

his model. Recall that the time allotment scheme of our country or economy, namely

equation (1.7) is

1 = u + ep + eq + l. (2.7)

As mentioned previously, l represents the time allotted for leisure throughout the

work day, u is the time allotted for production, ep is the amount of non-leisure time

spent in production education, or the human capital sector, and eq is the amount of

non-leisure, non-production time spent in health education. The summation of these

proportions is equal to one work day and 0 u, eq, ep, l 1. To recover Kejak or

Zilibotti, set l = eq = 0. Additionally, to fully recover Zilibotti, one would set ep = 0

as well.

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The equations above are important in answering economic questions and de-

termining how much time should be spent in each section of the economy. In order

to optimally allot time between the three sectors, we can assume that an individual

wants to maximize the combination of leisure and consumption over his/her lifetime.

This assumption is analyzed through examination of the maximization problem and

the differential equation for capital. The maximization problem uses the utility func-

tion from Ladron-De-Guevara et al. [9] and integrates over time

maxc,k,h,u,eq,ep,l

limQ

Q

0

[cl1]1

1

etdt . (2.8)In order to maximize the combination of leisure and consumption over a persons

lifetime, it is necessary to choose the optimal time allotments for u, eq, ep, l and also

for variables c, k and h. In this problem, c is consumption at time t, and is a

measure of risk aversion. Utility is the risk of gaining and losing money or resources.

For close to 0, a person/country is increasingly risk averse, avoiding risk whenever

possible, and current consumption is more important than future consumption. For

close to 1, a person/country is increasingly willing to take risk and considers current

consumption less important than future consumption and thus will save money to

use at a later time. This maximization problem is a trade-off between consumption

and leisure, and is the relative importance of consumption over leisure. For

close to 1, consumption becomes more important than leisure, and for close to 0,

leisure becomes more important than consumption. The time discounting factor is

represented by et, where is a factor that shows a consumers likelihood to invest

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at some point. A large implies people would rather spend their money now, and

consume all they earn when they earn it. A small , however, implies people are

willing to save their earnings and invest in capital or something else with future value

or benefit.

The equation for capital, which is a constraint on the maximization problem,

is

dk

dt= kr + wuh c, (2.9)

where r is the investment return rate, w is the working wage rate and k is capital.

This equation states that the derivative of capital with respect to time equals the

accumulating capital minus the capital spent. If a country does not consume all

that is earned, it will be available for capital. The w in this equation is found using

the profit maximization equation, which uses the Cobb-Douglas production function,

F(k, L) = GkL1, which models the production center. Here k is capital, L is labor,

G is a multi-productivity constant parameter and is a parameter that denotes the

capital share of production with 0 < < 1. Consequently, 1 corresponds to the

labor share of production. By setting profit equal to production minus wage, w, plus

Gross Domestic Product (GDP), we obtain the profit equation

= F(k, L) wL [DAk + Z(Ak)], (2.10)

which uses the production equation. Equation (2.10) will be optimized and the result

used to solve the associated Hamiltonian system of equation (2.8). The derivatives

with respect to capital and labor are taken to implement additional constraints on

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the parameters of production. The derivative with respect to labor lets us find w,

and the derivative with respect to capital obtains both D and Z. The derivations

for each are found in the Appendix. It is found that parameters D, Z and G are not

independent and maximizing profit yields the codependent relationship.

Another important assumption to note is

kr k y(t) = DAk + Z(Ak) , (2.11)

which states that capital, adjusted by its investment rate minus the depreciation of

capital, shown as k, is approximately equal to the Gross Domestic Product of the

economy [2]. This is used in (2.10). The first term, kr corresponds to earning by

investing and k corresponds to loss by depreciation. When they are subtracted, it

is the net rate of growth or decay of capital. This equation uses Zilibottis produc-

tyieldsion function, which has a linear and a non-linear term. The linear term is used

in Kejak also, where A is constant. The non-linear term is homogeneous and is added

so that a country is able to escape the trap of underdeveloped countries and provide

an opportunity for the fixed point to be unstable, so that it can initiate and maintain

growth. With the linear term alone, the fixed point is stable and countries are not

able to grow. Using Maple and, setting Z = 0, to eliminate the non-linear term, we

were able to verify that the fixed point was stable and there is little opportunity for

growth. Here, D and Z are parameters for an individual country and 0 1 is a

parameter associated with the non-linear factor of output. If we set = 0, then we

recover Zilibotti and similarly, setting = 1 recovers Kejak.

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2.2 Interpretation of Technology and Life Expectancy Equations

Recall that the interpretation of equation (2.2) in Section 2.1 is that as qK tends to

0, A tends to A0, which is the lowest level of technology. Also as qK increases, A

approaches a, the highest level of production technology. Figure 2.1 plots A versus

qK for different values of , the rate of diffusion of technology. As increases, the

economy more effectively utilizes resources to gain life expectancy and A approaches

a more quickly. However, if is too small, the economy is not effectively utilizing

resources, so it will not reach the highest level of technology very quickly, if at all.

Figure 2.1: Technology Equation (2.2): Varying , the rate of diffusion of technology.Here a = 0.075 and A0 = 0.015.

Equation (2.2) can represent any country, developed or developing, depending

on the range of qK. Each country would start at a different point on the curve, on

its level of capital, life expectancy and on the value of the diffusion parameter, the

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have a similar initial condition, or starting point, on a curve. As Figure 2.2 shows,

the increase in drastically increases the approach to Q, the highest life expectancy.

When = 0.03, thus more effectively utilizing capital or resources, the countrys life

expectancy starts at or near the highest life expectancy, Q. If individuals of a country

effectively utilize resources, the life expectancy of that country increases. However, if

individuals do not effectively use resources, life expectancy is lower, and the country

may not reach the highest life expectancy. Also, as eq increases, that is time spent

in health education increases, an economy reaches the highest life expectancy more

quickly.

0 1e

q

30

40

50

60

70

80

90

q

(in

years)

=3.0, =.001 & K=0=3.0, =.001 & K=100

=3.0, =.001 & K=200

=2.0, =.001 & K=100

=2.0, =.001 & K=200

Figure 2.3: Life Expectancy Equation (2.4): Varying , the measure of influence oftime spent in health education. Here Q = 85 and Q0 = 35.

Figure 2.3 demonstrates life expectancy influenced by the effectiveness of time

spent in education, . As with the previous case, the approach to the highest life

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expectancy is more rapid with larger . Increasing from two to three, implying that

the effectiveness of teachers teaching about the importance of health and teaching

about the risks of disease improves, adds several years to the life expectancy of that

country. When eq tends to 0, the average life expectancy approaches Q0, the feasible

minimum. As shown in Figure 2.3, even when K = 0, with an increase in the

amount of time spent in health education, eq, the country can approach the highest

life expectancy. Again, with the larger K, the approach to Q is faster for the two

values of .

0 1e

q

30

40

50

60

70

80

90

q

(in

years)

=3.0, =.03 & K=200

=3.0, =.03 & K=100

=2.0, =.001 & K=0

=2.0, =.001 & K=100

=2.0, =.001 & K=200

Figure 2.4: Life Expectancy Equation (2.4): Comparing developed versus developingcountries with varied , and K. Here Q = 85 and Q0 = 35.

Figure 2.4 compares rich and poor countries. A small or implies that the

country, rich or poor, is not very effective or mature with respect to using resources

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to improve health or health education. Countries that are not effective in using the

capital at hand or at teaching may not reach the highest life expectancy, regardless of

it being a rich or a poor country. The figure shows that, with small and small , the

country, whether rich or poor, does not reach the highest life expectancy, although the

life expectancy does increase. Rich countries have the advantage, however, regardless

of the values of or , because they start with a higher life expectancy than do poor

countries. Examining Figure 2.4, we notice that a country with K = 200 is very

effective in both using resources and teaching. Therefore, it begins at the highest life

expectancy and maintains that level. A country with K = 100 which is effective in

both using resources and teaching does not start at the highest life expectancy, but

does reach it. Therefore, it is not as important to begin with a lot of resources or

capital, it is more important to effectively utilize what you have.

2.3 Solving the Original Model

The Hamiltonian operator is used to maximize consumption over the variables c, k,

ep, eq, l, and h. We seek six governing equations for the six unknown variables.

The Hamiltonian is constructed by taking the utility function and adding to it two

constraints, one for human skill level and one for capital. Each contraint is multiplied

by a Lagrange multiplier, or . The Hamiltonian operator is

HAM(c, eq, ep, l , k , h; , ) =

(cl1)1

1 et

+ [kr + wuh c]

+ [B(H)eph] . (2.12)

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The Hamiltonian is rewritten by using equation (1.7) for u and adding a zero in the

form of +k k which yields

HAM =(cl1)1

1 e

t+ [(kr + k) k + w(1 ep eq l)h c] + [B(H)eph] . (2.13)

To continue solving the Hamiltonian, partial derivatives are taken with respect to the

eight variables and parameters, c, k, ep, eq, l, h, and , giving us eight equations

and eight unknowns, as follows:

HAMc = 0 = [cl1]l1c1et , (2.14)

HAM =dk

dt= (kr + k) k + w(1 ep eq l)h c, (2.15)

HAM =dh

dt= B(H)eph, (2.16)

HAMk =d

dt= [DA + ZAk1 ], (2.17)

HAMh = ddt = w(1 ep eq l) + B(H)ep, (2.18)

HAMep = 0 = B(H)h wh, (2.19)

HAMeq = 0 =

Dk

dA

dq

dq

deq+ ZkA1

dA

dq

dq

deq

wh, (2.20)

HAMl = 0 = c(1)(1 )l(1)(1)1et wh, (2.21)

where dAdq

and dqdeq

, which come from (2.2) and (2.4), are:

dA

dq=

aK(t)( aA0 1)e[qK]

[1 + ( aA0 1)e[qK]]2

, (2.22)

dq

deq=

Q( QQ0 1)e[(eq)+(K(t))]

[1 + ( QQ0 1)e[(eq)+(K(t))]]2

. (2.23)

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Now the above system of equations (2.14)-(2.21) can be solved by eliminating

and and reducing it to a system of six equations. We first, solve for using

(2.14) and use the Cobb-Douglas production function and the profit function to find

w. After such values are defined, we make the proper substitutions then continue

solving to find the governing equations. Appendix B has complete derivations to find

the following:

c

c=

DA + ZAk1

, (2.24)

hh

= B(H)ep, (2.25)

k

k= DA + ZAk1 + Gk1L1(1 )

c

k, (2.26)

d(DAk + Z(Ak))

deq= wh, (2.27)

deqdt

+ depdt

+ dldt

1 ep eq l=

h

h

k

k+

1

DA + ZAk1

1

(1 eq l)B(H) +

1

dB

dHeph, (2.28)

l =1

c

wh. (2.29)

Note when = 1, equation (2.29) yields l = 0 and leisure is not considered as impor-

tant as consumption. Maximization of (2.8) is equivalent to solving these equations

to determine the unknowns c,k,ep, eq, l and h.

2.4 Interpretations of the Original Model

Equation (2.24) means that the average rate of change of consumption is equal to

the marginal effect of capital on gross output, minus both the depreciation of capital

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and a time discounting factor, which is scaled by a relative risk-aversion coefficient,

. This equation shows that an increase in national output with respect to capital

results in an increase in average rate of consumption.

Equation (2.25) implies that the average rate of change of human skill level is

equal to the productivity scaled by production education. Interpreting this equation

suggests that more time spent in production education will produce workers with

higher skill levels, which will result in higher output [2]. Similarly, equation (2.27)

shows that the marginal effect of health education on production is equal to wage

times human skill level.

Equation (2.26), which is the capital governing equation, shows that capital is

equal to output and scaled production, less depreciation of capital and consumption.

As depreciation increases, the average rate of capital decreases, and this also decreases

as consumption increases. The average rate of change of capital increases as capital,

labor or technology increase.

Equation (2.28) shows that the average rate of change of time spent in pro-

duction is equal to the average rate of change of skill less the average rate of change

of capital plus scaled marginal effect of capital on output less depreciation of capital

and scaled productivity plus the rate of change of productivity scaled by time spent

in production. The implication is that time spent in education affects the entire

economic system.

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Equation (2.29) shows that leisure is equal to a scaled consumption divided

by a factor of a workers wage and the skill level of that worker. This shows that

when the wage is increased, leisure time decreases. Also, leisure time decreases when

the human skill level h, increases. This agrees with [1].

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CHAPTER III

STEADY STATE OF THE ORIGINAL MODEL

3.1 Steady State Solutions

The steady state solutions are found by setting the time derivatives of equations

(2.24) through (2.29) equal to zero. The equations and their solutions will be given

in this section, but the complete derivation process will be shown in Appendix C.

With these assumptions, the equations become

0 =DA + ZAk1

, (3.1)

0 = B(H)eph, (3.2)

0 = DA + ZAk1 + Gk1L1(1 ) c

k, (3.3)

d(DAk + Z(Ak))

deq= wh, (3.4)

0 =0

h

0

k+

1

DA + ZAk1

1

(1 eq l)B(H) +

1

dB

dHeph, (3.5)

l =1

c

wh. (3.6)

It is easiest to start with equation (3.2) to find a solution for ep. First, using the

productivity equation (2.5), this equation can be rewritten as follows:

dh

dt= B(H)eph =

BF

1 +

BFB0 1

eH

eph. (3.7)

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However, from above, the steady state solutions require time derivatives to be set to

zero. The only way dhdt

can equal zero is if h = 0, ep = 0 or B0 = 0. It is assumed

that individuals of a country have at least some skill level, so it wouldnt make sense

for the skill level h to be equal to zero. Setting B0 equal to zero would imply that

productivity never grows and that B(H) = dBdH

, which is not reasonable, so that is

eliminated as well. The only option remaining is ep = 0, which would imply that

there is no time spent is production education; therefore, the skill level h will not

increase. Now that we have ep set to zero and the time derivatives are equal to zero,

we look at the steady state solution for equation (3.5)

0 =1

DA + ZAk1

1

(1 eq l) B(H). (3.8)

Using equation (3.1), can be found, which will help solve for eq. We multiply both

sides by and add the to the other side to yield

DA + ZAk1 = . (3.9)

Now, equation (3.8) can be simplified as follows,

0 =1

1

(1 eq l) B(H). (3.10)

Following Kejak, it is assumed that B(H) is a constant and is a measure of produc-

tivity that is dependent upon h.

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We can finish solving for eq, holding B(H) constant. We next factor out1

in equation (3.10) to obtain

0 =

1

( (1 eq l)B(H)), (3.11)

then multiply both sides by to get

0 = (1 eq l)B(H). (3.12)

Next, we add the righthand side of the minus sign and divide by B(H) to obtain

1 eq l = B(H)

. (3.13)

Now, we finish solving for eq, which yields

eq = 1 l

B(H). (3.14)

Recall that 0 < eq < 1. Therefore, we must ensure that 0 (1 l

B(H)) 1.

This implies that 1 > l + B(H) > 0 and that B(H). This assumption can be

made because is a time discounting factor that is small and positive and B(H) is a

positive value but is larger than a discount factor. Since we have all positive values,

we are ensured that it is positive. It is important that is less than B(H), because if

is greater than B(H), the value would be too large and when added to l, would be

greater than one. Furthermore, l would have to be a value small enough that when

added to B(H)

, the value would not be greater than one.

We have solutions for ep, eq, and l, which is equation (3.6). Now we have to

solve for k, c, and h, where h is solved for numerically. We can get c and k in terms

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ofh using the Cobb-Douglas production function and the definition of labor, L = uh.

Using these equations and simplifying, as shown in Appendix C, yields the solutions

for k and c:

k =

B(H)h

G

+

1

1

, (3.15)

c =

B(H)h

G

+

11

(DA ) + G

G

+

1

(1 )

+Z(Ak). (3.16)

Using the derivatives of the profit function and algebraically manipulating

equation (2.20), yields the algebraic equation for h

dA

dq

dq

deq= A

1

B(H)

, (3.17)

which is solved numerically, as shown in Appendix C.

3.2 Results of the Original Model

Using both Maple and equations (3.6) and (3.17), we find eq and h. We are able

to use these values and equations (3.14)-(3.16) to find l, k and c. Technology A,

and life expectancy q, are found using equations (2.2) and (2.4). All of these values,

found in Table 3.1, are derived using specific values of parameters, which most are

taken from Zilibotti [5] and Kejak [6]. We have that = 0.08, = 0.05, G = 5, Z =

10, D = 1, = 0.72, B(H) = 0.0876, a = 0.075, A0 = 0.746x104, Q = 85, Q0 = 35

and = 0.4, where = 1.5 in the original model. We also have that = 0.00877/5.5

and = 0.01/5.5. These parameters are adjusted and scaled by 5.5 so that the values

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in the original model and transformed model match and a country will be at an equal

state on the curve for each model. For the steady state of the original model, we

assume that ep = 0. Looking at Table 3.1, we can see the trend as increases. Varying

is important, because it measures the value of leisure in a society. As expected, eq,

the time spent in health education, increases as increases. As increases, the time

spent in leisure decreases, therefore time spent in health education would increase.

Capital, k and consumption, c, decrease as increases. Consumption, therefore,

moves in the same direction as leisure, which validates equation (2.29) from Chapter

2. The human skill level h also decreases as increases, indicating the more leisure

time one has, the better his/her skill level becomes. Life expectancy q, however,

decreases with more leisure time. For close to 1, we get Tuckers results.

After we find the fixed point, we linearized the equations and solve the linear

equations using Maple. After solving these equations, which is a linear stability

analysis, we find the eigenvalues and eigenvectors of the fixed point. These are listed

in Table 3.2. Three eigenvalues corresponding to the fixed points are positive, which

indicate an unstable manifold. However, there is a negative eigenvalue, which is a

stable path leading to an economic trap. Should a developing countrys economy be

on a stable path, its economy will be trapped. Looking at the positive, real eigenvalue,

the eigenvector is pointing in the c direction.

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CHAPTER IV

CONSTRUCTING THE TRANSFORMED MODEL

4.1 The Transformed Model

A fixed point in the original model has a stable path which indicates a possible

economic trap, as was shown in Chapter 3. Since there is opportunity for an economic

trap, it is beneficial to perform a change of variables of the original model, creating a

transformed model. The transformed model will give a new set of fixed points which

are not fixed points of the original model. Note that ep = 0 in the original model, but

ep = 0 in the transformed model. We are considering a completely different model

whose fixed points describe dynamic paths in the original variables. We find in [9] a

change of variables to the model. We create variables for x and v:

x =k

h, (4.1)

v =c

k. (4.2)

These correspond to x being capital per human skill level and v the consumption

per unit of capital. Using these equations for x and v, we will seek a system of 5

equations with 5 unknowns: x,v,eq, ep and l. Our equations for technology (2.1) and

life expectancy (2.3) are similarly transformed.

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The transformation of equations (2.1) and (2.3) will set up the new 5-variable

system. Looking at the original technology equation (2.1), we see that

dA

dt =a A

a

A

q

dK

dt +

dq

dt K

. (4.3)

The technology equation becomes

A(t) =a

1 +

aA0 1

e[q(t)x(t)]

. (4.4)

Assuming k is constant as h increases, the level of technology would not ap-

proach a very quickly. This would imply that as human skill level increases, there isnt

as much of a need for new technology. It is assumed the individuals have mastered

the technology they currently have. However, k may not stay constant. Looking at

(2.15), k increases as h increases; therefore, increasing h could lead to an increase in

the level of technology, rather than a decrease. This increase in technology could be

a result of individuals requiring new technology to complement their enhanced skill

level. The parameter, , is the effectiveness of utilizing capital per human skill level

to gain life expectancy as technology increases. Equation (2.3), the life expectancy

equation, is also modified with a transformation of k to x as

dq

dt=

Q q

Qq

d

dt[eq + x] . (4.5)

The solution to this logistic equation is

q(t) =Q

1 +

QQ0 1

e[eq+x(t)]

. (4.6)

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With these transformations, we are now able to reduce our 6-variable system

of equations to a 5-variable system of equations. Considering the dependency of A

upon q, and q upon eq, equation (2.20) and its components fordAdq

and dqdeq

become

HAMeq = 0 =

Dk

dA

dq

dq

deq+ ZkA1

dA

dq

dq

deq

wh, (4.7)

dA

dq=

ax(t)( aA0 1)e[qx(t)]

[1 + ( aA0 1)e[qx]]2

, (4.8)

dq

deq=

Q( QQ0 1)e[eq+x(t)]

[1 + ( QQ0 1)e[eq+x(t)]]2

. (4.9)

Using equation (2.10) and taking the derivative with respect to capital, then setting

that equal to zero, ddk

= 0, we are able to solve for k. This k is used in equation

(4.7), as shown in Appendix C. Now, to find the first of our five equations used to

solve for our five unknowns, we use equation (3.17) and equation (3.13), which yields

dA

dq

dq

deq= A

1

1

1 ep eq l. (4.10)

Equation (3.13) is the steady state equation which assumes ep = 0, the above equation

does not make this assumption. Finding three of the next four equations is not quite

as straightforward and involves finding the derivatives and growth rates of x and v,

which are

dxdt

x=

dkdt

h dhdt

k

h2h

k=

dkdt

k

dhdt

h, (4.11)

dvdt

v=

dcdt k dkdt ck2

kc

=dcdt

c

dkdt

k. (4.12)

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Now, substitute equations (2.24), (2.25) and (2.26) into equations (4.11) and

(4.12) and transform the appropriate variables, when needed, to obtain three of the

next four equations:

x

x= DA

1

1

+ G

1 ep eq l

x

1 1 +

v B(H)ep, (4.13)

v

v= G

1 ep eq l

x

1 1 +

+ v +

1

1

DA1

1

, (4.14)deqdt

+ depdt

+ dldt

1 ep eq l=

dxdt

x+

1

G

1 ep eq l

x

1

1

(1 eq l) B(H). (4.15)

The last equation that will be used to solve for the five unknowns is found by using

equation (2.29), performing a multiplication and division by k and transforming the

appropriate variables, to yield

l =(1 )xv

1epeql

x

G(1 )

. (4.16)

We can now use (4.10), (4.13), (4.14), (4.15) and (4.16) to solve for our fixed points

x,v,ep, eq and l.

4.2 Finding the Fixed Points

In order to solve for x,v,ep, eq and l, it is necessary to manipulate (4.10) and (4.13)-

(4.16) for use in Maple. First, we use equation (4.10) and equations (4.8) and (4.9)

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to find ep and use a simple division in equation (4.16) to find v. We also, isolate

deqdt

in equation (4.15) to find equation (4.22). The complete derivations are found in

Appendix C. The following equations are used in Maple:

ep = 1 eq l

1

1 + ( a

A0 1)e[qx]

1 + ( Q

Q0 1)e[eq][x]

2x

aA0 1

e[qx]Q

QQ0 1

e[eq][x]

, (4.17)

v =lG(1 )x

(1 )x(1 ep eq l), (4.18)

dxdt

x= DA

1

1

+ G

1 ep eq l

x

1 1 +

v B(H)ep, (4.19)

dv

dt

v= G

1 ep eq l

x

1 1 +

+ v

+

1

1

DA

1

1

, (4.20)

deqdt

+ depdt

+ dldt

1 ep eq l=

dxdt

x+ G

1 ep eq l

x

1

B(H)

(1 eq l), (4.21)

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deqdt

=

(1 ep eq l)

dxdt

x+ G

1epeql

x

1

1 + dep

deq

(1 ep eq l)

B(H) (1 eq l)

depdx

dxdt

1 + depdeq

, (4.22)

Since eq, x and l depend on time, we use the chain rule to calculatedepdt

.

depdt

=depdeq

deqdt

+depdx

dx

dt+

depdl

dl

dt, (4.23)

where depdl

= 1 and depdeq

and depdx

are as follows:

depdeq

= 1 + 1

x( aA0 1)Q( Q

Q0 1)

dTdeq

B dBdeq

T

B2

, (4.24)

T =

1 +

a

A0 1

e[qx]

1 +

Q

Q0 1

e[eq][x]

2,

dT

deq=

a

A0 1

e[qx]x

dq

deq

1 +

Q

Q0 1

e[eq][x]

2

21 + QQ0

1 e[eq][x] QQ0

1 e[eq ][x]

1 +

a

A0 1

e[qx]

,

B = e[qx][eq][x],

where B is with respect to eq and is replaced in equation (4.24) and

dB

deq= e[qx] x

dq

deqe[eq][x],

depdx

= 1

( aA0 1)Q( Q

Q0 1)

dTdx

B dBdx

T

B2

, (4.25)

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dT

dx=

a

A0 1

e[qx]

x

dq

dx+ q

1 +

Q

Q0 1

e[eq][x]

2

2

1 +

Q

Q0 1

e[eq][x]

Q

Q0 1

e[eq][x]

1 + a

A0 1

e[qx]

,

B = xe[qx][eq][x],

where B is with respect to x and is replaced in equation (4.25) and finally

dB

dx= e[qx][eq][x]

1 x

+

dq

dxx + q

.

We still need dldt , which requires the use of the chain rule on (4.18) because of

the time dependent nature of l ,x,ep and eq as follows:

dv

dt=

dv

dl

dl

dt+

dv

dx

dx

dt+

dv

dep

depdt

+dv

deq

deqdt

. (4.26)

Rearranging and making the substitution for depdt

to solve for dldt

we obtain

dldt =

dv

dt

dv

dx

dx

dt

dv

deq

deq

dt

dv

dep

dep

deq

deq

dt

dv

dep

dep

dx

dx

dtdvdl

+ dvdep

depdl

, (4.27)

where

dv

dl=

G(1 )x1

1

1 + l(1 ep eq l)

1

(1 ep eq l)

, (4.28)

dv

dx=

G(1 )( 1)x

(1 )(1 ep eq l)x2, (4.29)

dv

dep=

dv

deq=

G(1 )lx1

(1 )(1 ep eq l)+1. (4.30)

Notice that dvdep

and dvdeq

are equivalent.

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4.3 Results of the Transformed Model

In Chapter 2 we assumed a starting technology and life expectancy level when in-

terpreting equations (2.2) and (2.4). After changing variables, it is appropriate to

assume that Q0 = 35, however A0 needs to be much smaller, because we want A

to have a similar curve and initial position on the curve for the transformed model.

We compare Figure 4.1 with Figure 2.1 to see that adjustments were made with our

values of A0 and so that the two figures will have curves that start at the same

initial position. Our goal is to use the fixed points of c, k and h to find similar fixed

points in x and v.

0 10 20 30 40 50q

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

A

=.0093 & x=18.205

Figure 4.1: a = 0.075 and A0 = 0.1404x103.

Figure 4.1 shows the new value of A0 = 0.1404x103, and the approach to

the highest level of technology as q increases. Figure 4.2 shows the approach to the

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highest life expectancy as eq increases. For a country to reach a life expectancy of

Q, the time spent in health education, eq, has to approach 1. It is unlikely that a

country would spend its entire work day learning health education. Therefore, it is

not likely that a countrys life expectancy will reach 85, according to this model.

0 1e

q

40

50

60

70

80

90

q

(in

year

s)

=4.0, =.02 & x=18.205

Figure 4.2: Q = 85 and Q0 = 35.

Again, using both Maple and equations (4.19), (4.22) and (4.27), we find x, eq

and l. Using these values and equations (4.17) and (4.18), we are able to find ep and

v. These fixed points are dynamic in h, k and c. We use equations (4.4) and (4.6)

to find A and q. The parameters used to find the fixed points in the transformed

model are: = 0.08, = 0.05, G = 5, Z = 10, D = 1, = 0.72, B(H) = 0.0876, a =

0.075, A0 = 0.1404x103, Q = 85, Q0 = 35, = 0.4, = 4, = 0.00932 and = 0.02.

Since the size of determines the amount of time spent in leisure, simulations were

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education does not play a role in leisure time. However, the effectiveness of health

education is important for ep and eq. Notice that when = 3, the graph for ep is

higher than when = 4. If a country is not effectively teaching health education,

the individuals will seek education elsewhere. On the other hand, the values of eq for

= 4 is greater than that for = 3, because a country would spend more time in

health education if it is being taught properly. The sum of the three variables used

in Figure 4.3 is plotted versus also to show that the increase in ep and eq is greater

than the decrease in l, hence the graph increases as tends to 1.

0.94 0.95 0.96 0.97 0.98 0.99 1

0.84

0.85

0.86

0.87

0.88

0.89

0.9

0.91

u

=3.0

=4.0

Figure 4.4: Time spent in production, u, versus with two values of where Q0 =35, = 0.00932 and = 0.02.

Looking at Figure 4.4, as tends to 1, the time spent in production, u,

decreases. Recall that, as increases, the amount of time spent in leisure decreases.

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This indicates that as leisure time increases slightly, the production u, increases. The

result is similar for both = 3 and = 4. In this case, a larger increases u,

indicating more effective teaching leaves more time for production.

0.94 0.95 0.96 0.97 0.98 0.99 1

42

43

44

45

46

47

q

(in

years)

=3.0

=4.0

Figure 4.5: Life expectancy, q, versus with two values of where Q0 = 35, =0.00932 and = 0.02.

Although productivity increases as leisure time increases, the result is not

the same for life expectancy q. Figure 4.5 shows that q increases as increases.

Therefore, the less time spent in leisure, the longer the life expectancy. One may be

more productive throughout his/her life with having more time for leisure, but that

persons life expectancy decreases as a result. Notice that more effectively teaching

about health ( increasing), increases life expectancy.

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As shown in Figure 4.6, the technology level for a country decreases as tends

to 1. This indicates that as leisure decreases, a countrys technology level decreases

also. More effectively teaching health also increases the level of technology.

Figure 4.6: Technology, A, versus with two values of where Q0 = 35, = 0.00932and = 0.02.

Recall that v = ck

, which is consumption per level of capital. In Figure 4.7,

this ratio increases with the increase in . With less time available for leisure, v

increases. On the other hand, Figure 4.8 shows that x decreases as increases. Also

recall that x = kh

, so that as leisure decreases so does the ratio of capital per human

skill level.

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0.94 0.95 0.96 0.97 0.98 0.99 1

0.5

0.502

0.504

0.506

0.508

0.51

v

=3.0

=4.0

Figure 4.7: Ratio of consumption per capital, v, versus with two values of whereQ0 = 35, = 0.00932 and = 0.02.

A stability analysis for the fixed points is important to determine if a country

will reach a better state of living. The parameters used for the stability analysis are

the same as are used when finding the fixed points, which are: = 0.08, = 0.05, G =

5, Z = 10, D = 1, = 0.72, B(H) = 0.0876, a = 0.075, A0 = 0.1404x103, Q =

85, Q0 = 35, = 0.4, = 4, = 0.00932 and = 0.02.

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0.94 0.95 0.96 0.97 0.98 0.99 1

16

16.5

17

17.5

18

18.5

19

x

=3.0

=4.0

Figure 4.8: Ratio of capital per human skill level, x, versus with two values of where Q0 = 35, = 0.00932 and = 0.02.

Table 4.1 shows there are two positive eigenvalues and one negative eigen-

value for each used. The two positive eigenvalues indicate instability and change in

the state of an economy. This change is for the better and is improving the state of

the economy. Negative eigenvalues, however, indicate a trap. The countrys economy

is trapped at a certain state, although it is at a better place than before, perhaps

with improved capital or technology. If a country happens to get trapped, changing

will improve the economic state at which the country is trapped. Notice that as

increases, there is not a significant change with the positive eigenvalues, but the

negative eigenvalue decreases. The same analysis was performed with = 3 to see

if there was significant change in the eigenvalues. For = 3, the positive eigenval-

ues did not have a significant change when varying , but the negative eigenvalue

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decreased as increased. The positive eigenvalues for = 3 and = 4 did not

show significant change, while the negative eigenvalues were smaller for the smaller

. The eigenvectors corresponding to each eigenvalue are also given showing that

for the positive eigenvalues, the eigenvectors point in the x direction in the (eq, x , l)

coordinate system.

Table 4.1: Eigenvalues and Eigenvectors for the Transformed Model

Eigenvalues Eigenvectors

eq, x, and l

-0.5288 0.0948, -3.1736, -0.0044

= 0.95 0.5164 -0.9432, 41.7519, 0.0303

0.1159 -0.7551, 34.0059, 0.1270

eq, x, and l

-0.5408 0.9543, -29.8633, -0.0080

= 0.99 0.5188 -8.4528, 358.7516, 0.0491

0.1171 -4.1906, 182.4620, 0.1345

Figures 4.9-4.11 are direct numerical solutions of equations (4.17)-(4.19),

(4.22) and (4.27). Using the linear stability analysis, a small factor, of order 1012, is

added to the fixed point to determine an initial condition, which is as close as possible

to stability. Adding such a small factor does not drastically alter the fixed point and

looking at Figures 4.9-4.11, when time is equal to zero, we can see that the initial

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0 10 20 30 40 50

time (in years)

0

0.01

0.02

0.03

0.04

0.05

0.06

l

=.95

=.99

Figure 4.10: Time spent in leisure, l, versus time, in years to see if the stable pathcan move away from the fixed point.

0 10 20 30 40 50time (in years)

16

18

20

22

24

x

=.95

=.99

Figure 4.11: Ratio of capital per human skill level, x, versus time, in years to see ifthe stable path can move away from the fixed point.

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CHAPTER V

CONCLUSIONS

What can be done to help developing countries grow? Developing countries are

stagnant in a bad economic state, and the goal is to discover if there is anything that

can help these countries reach a better one. Various models were studied to find the

best fit one for a developing country. We started with the models of Jones and Romer,

but their models reflect a developed country more than a developing country. These

two led into Zilibotti and Kejak, whose models better reflect a developing country.

Tucker added life expectancy into his model, which incorporated health ed-

ucation in the work day. He found that adding health education does increase life

expectancy and thus does improve the economic state of a developing country. A

country can still be trapped, but will be trapped at a better place. In addition to

health education, we incorporate leisure and try to determine if this modification

helps an economy reach a higher economic state.

After solving the original variable model and doing a stability analysis, we

deduced that the country can get trapped. Studying the transformed model was more

promising. Although a country can still be trapped, the trap can occur at a better

economic state. It is not determined whether or not a developing country actually

grows. It would be necessary to study per-capita income of a country to determine

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economic growth. For the purpose of this model, it has been shown that incorporating

health education and leisure improves the economic state of a developing country.

Further research is necessary to get a clearer understanding of economic

growth for a developing country. Including a stress factor into the life expectancy

equation would be a good idea for continuing this research. Another good idea to

continue this research would be to break ones life into stages to see when someone is

most productive and how that affects the economic state of a country. Using a model

that incorporates per-capita income would provide an answer to the question of what

can help developing countries grow, which is also a good idea for future research.

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BIBLIOGRAPHY

[1] D. Conley. Rich mans burden. The New York Times, September 2008.

[2] J. Tucker. A three-sector, integrated approach to economic growth modeling:Production, human capital, and health education. Masters thesis, The Univer-sity of Akron, Akron, OH, 2008. unpublished.

[3] C.I. Jones. Introduction to Economic Growth. W. W. Norton and Company Inc,

Second edition, 1998.

[4] D. Romer. Advanced Macroeconomics. McGraw-Hill Companies Inc, Third edi-tion, 1996.

[5] F. Zilibotti. A Rostovian model of endogenous growth and underdevelopmenttraps. European Economic Review, 39:15691602, 1995.

[6] M. Kejak. Stages of growth in economic development. Journal of EconomicDynamics and Control, 27:771800, March 2003.

[7] R. Lucas Jr. On the mechanics of economic development. Journal of MonetaryEconomics, 22:342, July 1988.

[8] D. Acemoglu and F. Zilibotti. Productivity differences. The Quarterly Journalof Economics, 116:563606, 2001.

[9] A. Ladron-De-Guerva, S. Ortigueira, and M. Santos. A two-sector model ofendogenous growth with leisure. Review of Economic Studies, 66:609631, April1998.

[10] P. A. de Hek. An aggregative model of capital accumulation with leisure-dependent utility. Journal of Economic Dynamics and Control, 23:255276,1998.

[11] E. G. Dolan and D. E. Lindsey. Introduction to Microeconomics. Best ValueTextbooks, LLC, First edition, 2004.

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[12] S. Ortigueira. A dynamic analysis of an endogenous growth model with leisure.Economic Theory, 16:4362, 2000.

[13] C. Vellutini. Capital mobility and underdevelopment traps. Journal of Develop-ment Economics, 71:435462, 2003.

[14] R. Aisa and F. Pueyo. Endogenous longevity, health and economic growth: aslow growth for a longer life? Economics Bulletin, 9:110, 2004.

[15] A. van Zon and J. Muysken. Health and endogenous growth. Journal of HealthEconomics, 20:169185, 2001.

[16] C. A. Echevarria and A. Iza. Life expectancy, human capital, social security andgrowth. Journal of Public Economics, 90:23232349, 2006.

[17] C. Arndt. HIV/AIDS, human capital, and economic growth prospects for Mozam-bique. Journal of Policy Modeling, 28:477489, July 2006.

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APPENDICES

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APPENDIX A

RECOVERING TUCKER

Fabrizio Zilibottis and Michal Kejaks models played an important role in defining

this model. Their models can be recovered by substituting a 1 or 0 for certain

variables, which can be seen in the Appendix of Tucker [2]. For the purpose of this

model, Tuckers model will be recovered. Showing that his model can be recovered

indicates that Zilibotti and Kejak will also be recovered when following the steps

Tucker uses.

In order to recover Tuckers equations, set = 1 and l = 0. l directly relates

to leisure and comes from the utility function that incorporates leisure into the

model. Therefore, a good place to start is with the maximization problem, which

comes from the utility function. In (2.8), setting = 1 and l = 0, will yield Tuckers

maximization problem, which is

maxc,k,h,u,eq,ep

lim

Q

Q0

c1

1 etdt

. (A.1)

To recover Tuckers formula for the time allotment scheme for a country or economy

in a given work day, it is necessary to set l equal to zero in (2.7),

1 = u + ep + eq. (A.2)

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Looking at the Hamiltonian operator, (2.13), with = 1 and l = 0 we have:

HAM =

c1t

1 et

+ [(kr + k) k + w(1 ep eq)h c]

+ [B(H)eph] . (A.3)

Again, for completeness, the remaining equations will be manipulated to recover

Tuckers equations. Starting with (2.14), set the parameter, , equal to one, to find

HAMc = 0 = cet . (A.4)

Equations (2.16),(2.17),(2.19) and (2.20), match Tuckers equations exactly. Also,

(2.21), the derivative with respect to l, would not be in this model if setting l = 0,

removing one equation from the system. Therefore, this model would not have an

additional equation which Tucker does not have. To finish recovering Tuckers model,

in (2.15) and (2.18) we set l = 0 to find

HAM = dkdt

= (kr + k) k + w(1 ep eq)h c, (A.5)

HAMh =d

dt= w(1 ep eq) + B(H)ep. (A.6)

Making these adjustments recovers Tuckers model completely.

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APPENDIX B

HAMILTONIAN DERIVATIONS

The Hamiltonian and its governing equations are presented in Chapter 2, but the

derivations of such equations were not provided. In this section, the derivations of

the governing equations will be shown.

B.1 Consumption Governing Equation

Use (2.14) to solve for and find

[cl1]l1c1et = . (B.1)

This can be substituted into (2.17) to obtain

[[(l1c1et)((cl1)1)(c(1 )l1dl

dt+ l1c1

dc

dt]

+[(cl1)(c1et(1 )ldl

dt+ l1et( 1)c

dc

dt

+l1c1et )]] = [cl1]l1c1et[DA + ZAk1 ], (B.2)

which simplifies to

lc

c1c + lc

= lc

[DA + ZAk1 ]. (B.3)

Finally, after dividing through by lc

, one can manipulate this to obtain (2.24):

c

c=

DA + ZAk1

. (B.4)

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B.2 Capital Governing Equation

Before showing the derivation of the Capital Governing Equation (2.26), we explain

the derivation of (2.25). Using (2.16), a simple division will yield (2.25). However, the

derivation for (2.26) is not so straightforward. In order to solve for this, a one sector

production firm in the economy will be assumed through a Cobb-Douglas production

function. As mentioned earlier, the Cobb-Douglas production function is:

F(k, L) = GkL1, (B.5)

with profit,

= F(k, L) wL [DAk + Z(Ak)]. (B.6)

In order to maximize profits, the following condition must be met:

d

dL= 0 = FL w. (B.7)

For completeness, the derivative with capital is also taken, which is

d

dk= 0 = Fk DA ZA

k1. (B.8)

This maximization adds another constraint on ep, eq,h,A,k,D and Z.

Now that the background has been set, the solution for the Capital Governing

Equation can be shown. First, we divide (2.15) through by k, keeping in mind that

k = K, to yield

k

k= DA + ZAk1 +

w(1 eq ep l)h

k

c

k. (B.9)

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At this time, we introduce the labor equality:

L = uh = (1 eq ep l)h. (B.10)

Therefore, (B.9) can be rewritten as

k

k= DA + ZAk1 +

wL

k

c

k. (B.11)

We solve for w using (B.7) to yield

w = FL = GkL(1 ). (B.12)

Now, (B.11) above can be rewritten as

k

k= DA + ZAk1 +

GkL(1 )L

k

c

k. (B.13)

The final governing equation is

k

k

= DA + ZAk1 + Gk1L1(1 ) c

k

. (B.14)

B.3 Health Education Governing Equation

Equation (2.20), with a simple division by , can be written as

wh =

Dk

dA

dq

dq

deq+ ZkA1

dA

dq

dq

deq

. (B.15)

From these two, we looked at models that include leisure. Notice each term on the

right side has a common factor, so that

wh =dA

dq

dq

deq

Dk + ZkA1

. (B.16)

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This equation can be reduced to

wh =d(DAk + Z(Ak))

deq. (B.17)

It is seen through this equation that one skilled work hour is equal to the marginal

effect of health education on production.

B.4 Production Education Governing Equation

The solution for this equation is a little more involved and tedious than the previous

equations, and starts with (2.19) by rewriting it as

B(H) = w. (B.18)

This is substituted into (2.18) to obtain

d

dt= B(H)(1 ep eq l) + B(H)ep. (B.19)

This can be simplified to

d

dt= B(H)(1 eq l). (B.20)

Starting with (2.19), dividing throughout by h, and taking the derivative with respect

to time yields

d

dt

w +dw

dt

+d

dt

B(H) B(H)dh

dt

= 0. (B.21)

Using (B.19) and (2.16) and solving for we find that = wB(H)

, and

d

dtw +

dw

dt + B(H)(1 eq l)B(H) B

(H)B(H)ephw

B(H)= 0. (B.22)

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Using the above equation, we simplify and divide by w to obtain

ddt

+

dwdt

w+ (1 eq l)B(H) B

(H)eph = 0. (B.23)

We can use (2.17) to make further simplifications as follows,

dwdt

w

DA + ZAk1

+ (1 eq l)B(H)B(H)eph = 0. (B.24)

We next solve fordwdt

wby taking a derivative with respect to time in (B.12), which

yields

dw

dt = G(1 )(k1

kL

L1

Lk

). (B.25)

Divide through by w,

dwdt

w=

G(1 )kL(k1k L1L)

G(1 )kL. (B.26)

This can be simplified further as

dw

dtw =

k

k L

L, (B.27)

where = ddt

.

We know kk

which is given (2.26), but we have to solve for LL

using (B.9):

L

L=

d((1epeql)h)

dt

(1 ep eq l)h=

dhdt

(1 ep eq l) (depdt

+ deqdt

+ dldt

)h

(1 ep eq l)h. (B.28)

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This equation is substituted back into the above equations and simplifications

are made to yield (2.28)

deqdt

+ depdt

+ dldt

1 ep eq l =

h

h

k

k +

1

DA + ZA

k1

1

(1 eq l)B(H) +

1

dB

dHeph. (B.29)

B.5 Leisure Time Governing Equation

The final governing equation, dealing with leisure, is solved in a straightforward

fashion. Using (2.21), we rewrite as follows:

c(1)(1 )l(1)(1)1et = wh. (B.30)

Substituting the known value of , using (B.1), we obtain

[cl1]l1c1etwh = c(1)(1 )l(1)(1)1et. (B.31)

This equation can be simplified to

1

clwh = 1 . (B.32)

Solving for l yields (2.29),

l =1

c

wh. (B.33)

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B.6 Transforming the Governing Equations

The derivations for finding the governin

rollo sara k

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